Returning to Princeton University, Liu Yichen began to gradually adjust his condition.

In the library, Liu Yichen was spinning his pen in his hand and his mind was racing rapidly.

As the saying goes, combining work and rest, I spent more than three months in the summer vacation and had a good month of fun. I didn't think about research in other worlds. At this time, I adjusted my attention to mathematics, and inspiration continued to emerge.

"Maybe group theory can solve the problems I'm currently encountering very well!" Liu Yichen thought to himself.

Group theory is a very powerful tool. Not only is it one of the two major theoretical artifacts of quantum mechanics along with Hilbert space in functional analysis, it can often be used in number theory, especially when studying infinite prime number problems.

Wonderful effect.

For example, any teacher of basic number theory will mention a very classic example in the first or second class - Fermat's Little Theorem.

There are many ways to prove this theorem, among which the most concise proof method is recognized to be proved by group theory.

As for how simple it is, standard fonts can even be achieved with just three lines.

That is, if α and p are relatively prime, according to Euler’s theorem, α^φ(p)≡1(modp), but φ(p)=p-1, so α^(p-1)≡1(modp), both sides

Multiplying α can lead to the conclusion: when α is a natural number and p is a prime number, α^p≡α(modp).

Is not it simple?

In fact, Fermat's little theorem is just a special case of Euler's theorem.

However, using Euler's theorem, it can still be solved using group theory methods, and the entire proof process does not take half a page.

He had previously proved the twin prime conjecture. When Liu Yichen was thinking about the proof of the Polignac conjecture, he had been thinking about how to extend the k=1 form to infinite natural numbers. He first thought of carrying out the topology principle of the sieve method.

Supplement, but encountered obstacles.

Now that he has been relaxing for more than three months during the summer vacation, when Liu Yichen thought about it again, he thought of group theory.

Inspired, Liu Yichen began to write down rows of calculations on the scratch paper.

After several days, Liu Yichen finally completed the proof of Polignac's conjecture. Looking at the dozen or so pages of draft paper he had written, Liu Yichen showed a faint smile.

During this time, he saw many papers on the arxiv website, all of which wanted to further prove the Polignac conjecture on his conjecture of twin prime numbers. However, there is no doubt that those have fatal thinking errors.

Some of them may be that the author did not discover them, while others may be that the author wanted to avoid them by taking advantage of them.

But for Liu Yichen, whose research on number theory has reached the level of a master, this cannot escape his eyes.

"The next step is to sort out the papers!" Liu Yichen did not leave the door, so he sorted out the papers in the dormitory. Feng Lin brought him three meals.

After completing the proof of the Polignac conjecture, it will be much easier to compile the paper. Liu Yichen first compiled the Chinese version of the paper, which has more than ten pages of draft paper, but the final paper has 48 pages. And the English version is enough.

At 63 pages, it is so long that it can be divided into two papers.

However, Liu Yichen did not divide it into two papers. Unlike the last time he proved the weak Goldbach conjecture, this time he proved the Polignac conjecture, which is consistent from top to bottom.

After saving the completed paper, he uploaded it to the arxiv website and then submitted it to the "Annual Review of Mathematics". Liu Yichen couldn't help but stand up and stretch, making a crackling sound all over his body.

When he walked to the window and opened the curtains, he felt that the air had become fresher, and Liu Yichen felt indescribably happy.

Not just because he solved another mathematical problem and got one more sci paper, but because solving this mathematical problem gave him a deeper understanding of group theory, and on this basis he developed a set of

Brand new mathematical methods.

And this discovery made him even more excited than solving the mathematical conjecture itself.

Hilbert once commented that Fermat's Last Theorem is a chicken that can lay golden eggs, not because this hen has fed a large number of mathematicians, nor because this hen has provided many journals with opportunities for water papers.

, but because many novel mathematical methods are derived from the study of number theory problems.

For example, inspired by Fermat's problem, Coomer introduced the concept of ideal numbers and discovered the unique decomposition theorem that decomposes a number in a cyclic field into ideal prime factors. This theorem has been popularized by Dedekind and Kronecker today.

To any algebraic domain, it occupies a central position in modern number theory, and its significance has gone far beyond the scope of number theory and penetrated into the field of algebraic function theory.

The significance of studying mathematical conjectures is not that solving a mathematical conjecture makes it become a theorem. This is actually not important. What is really important is the new mathematical tools produced in the process of studying mathematical conjectures, or new insights into existing mathematical tools.

Use or improve existing mathematical tools to promote the development of mathematics.

"Perhaps the previous understanding of the mathematical community is reasonable. If you want to solve the '11' form of Goldbach's conjecture, you must find a new method." Liu Yichen thought to himself.

When Chen Jingrun proved Goldbach's conjecture '12', it can be said that he applied the sieve method theory to the extreme. It is precisely because of this that under the sieve method theory, no one has been able to surpass Chen Jingrun's '12' in the past half century.

research results.

After taking a shower, tidying up, and changing into a set of clean clothes, Liu Yichen looked at the time. At almost four o'clock in the afternoon, Liu Yichen went to Professor Deligne's office.

"Hey, Liu, why did you come to the office? Does the sun rise from the west today?" Bella was quite surprised. This person is a legend. He entered Professor Deligne's door with a lot of honors and didn't need to go to class.

When I come to the office every day, I don’t even need to participate in the research, I just do my own research.

Liu Yichen didn't come to the office many times.

"Bella, you haven't graduated yet, I thought you had already obtained a doctorate!" Liu Yichen said strangely when he saw Bella.

"Stop talking, my doctoral thesis was rejected by the professor. It seems that I have to work hard for another year or even two years!" Bella couldn't help but become frustrated when she heard this.

"Liu, you're here!" Professor Deligne couldn't help but smile with relief when he saw Liu Yichen.

This year, Liu Yichen has made another achievement in number theory, solving the weak Goldbach conjecture and the twin prime conjecture. These two mathematical conjectures are quite significant.

"Hi, Professor!" Liu Yichen greeted Professor Deligne with a smile, and then placed the tea in his hand on Professor Deligne's table: "This is tea from my hometown. My hometown is the hometown of tea. I brought it to you specially.

I ordered a cup of tea, but it’s a pity that you weren’t there when I returned to school.”

"Thank you, your tea is the best gift for me!" Professor Deligne said: "How is your research going recently?"

"I proved the Polignac conjecture on the basis of proving the twin prime conjecture. The paper has just been submitted to the "Annals of Mathematics"!" Liu Yichen said happily: "I think it won't take long before I can complete Goethe's conjecture.

The proof of Bach's conjecture '11' has completely capped the building of Goldbach's conjecture!"

When Bella heard Liu Yichen's words, she almost spat out blood. She studied under Professor Deligne for four years, but failed to even pass her doctoral thesis. However, Liu Yichen did better, publishing one SCI paper after another, and it was published in "Mathematics"

Annual Papers.

This means that so far, Liu Yichen has solved three world-wide mathematical problems within one year, and he seems to want to prove Goldbach's conjecture in one go.

Damn it, can you stop being such a monster!

At this moment, Bella felt very hurt. She thought that she was also an IMO gold medal winner, and was recommended to Princeton University. She studied mathematics at Princeton University for four years, and then successfully entered Professor Deligne. She thought she could also successfully obtain

After getting a doctorate degree, she saw the gap in these four years.

Even though the other two had studied for one year less than him, their level was already above hers.

Not to mention Liu Yichen, who only entered Prince University for one year and already left her far behind. She couldn't even see Liu Yichen's back.

"So fast?" Professor Deligne was quite surprised.

"How about it, professor, can it meet your requirements?" Liu Yichen said with a smile.

"Haha, if you prove Goldbach's conjecture, I can give you the nod to get a doctorate!" Professor Deligne said with a chuckle: "Print out your paper that proves Polignac's conjecture and show it to me.

look!"

Liu Yichen turned on the computer, logged in to his account, printed out the paper, and then handed the paper to Professor Deligne.

Professor Deligne took the paper and began to read it seriously.

"What a pervert!" Bella looked at Liu Yichen and muttered to herself, then logged on to the arxiv website and saw Liu Yichen's paper proving Polignac's conjecture.

Bella commented: "He has proved the Polignac conjecture and is launching a charge thinking about Goldbach's conjecture. You don't need to study the Polignac conjecture and Goldbach's conjecture. You can't compare with him!"

Soon, someone replied on the comment: "He proved the Polignac conjecture? Damn it, I was just a little bit close to proving it! He is a world-class mathematician. In the world of number theory,

Big talk, why don't you leave some soup for us to drink and kill them all like this?"

"I have been studying Goldbach's conjecture for ten years, and he is so cruel to attack Goldbach's conjecture!" After a few minutes, someone else replied in the comments, expressing extremely angry.

Of course, the more important thing is to download the paper, and then conduct research to see if there are any problems in the paper. If there are problems, but there are lessons to be learned, then you can be the first to complete the complete paper, and the honor belongs to you.
Chapter completed!